The present invention relates to a method of producing impact point position probability maps for a well over a surface included within a three-dimensional medium and the position of each point of which is known with geometrical uncertainties.
The geometrical position uncertainties are vectors whose components are expressed in units of length, for example in metres, in the three dimensions of the medium.
To ascertain the geometrical structure of the subsurface, it is usual, in oil exploration in particular, to construct what is referred to as a 3D seismic block and the corresponding set of stack velocities. The means for obtaining a 3D seismic block are well known to experts and will not be described in detail. Having obtained a satisfactory seismic block, a migration step is generally carried out. The purpose of the migration operation is to provide the interpreter with an image of the subsurface which is as near as possible to a geological slice.
Two types of migration are chiefly used, namely time migration and depth migration. Time migration provides an acoustic image. Depth migration provides as exact as possible a depthwise image of the subsurface. However, the two types of migration require the prior definition of a velocity model, which can be obtained, for example as proposed by LANDA (Geophysical Prospecting No. 86, pp. 223-243, 1988) or else by WHITCOMBE (61st SEG, Houston, 1991).
In practice, the velocity models are marred by uncertainties, this rendering the results of the migrations inaccurate by the same token. In an article presented at the EAEG 1996 (Amsterdam Jun. 3-4 1996), P.THORE and A.HAAS proposed a practical and simple method for determining the migration errors due to the uncertainty in the velocities, as well as simplified formulae which can be used depending on whether depth migration or time migration is present. These formulae may be regarded as a first-order approximation for determining the direction and modulus of an uncertainty vector in the case of a variable velocity field and provided that one is within the domain of validity of the DIX formula, that is to say with uniform lateral variations in velocities in respect of plane and moderately-sloping reflectors.
In the patent FR-A-2 763 702, there is described a method of producing risk maps for the positioning of a well within a medium as a function of migration errors due to the uncertainty in the velocity field. The object of the method is to perform as many migrations of a seismic horizon as one has possible velocities and to tick off the points of impact of a fixed drilling well on each occurrence of location of the migrated horizon.
Moreover, the determination of a trajectory of a drilling well depends, on the one hand, on the choice of a departure point and, on the other hand, on the choice of a target point to be reached in the subsurface. Conventionally, the target point lies on a surface present in the 3D seismic block, the said surface constituting a seismic horizon. Given that the locations of the seismic horizons are affected by geometrical uncertainty vectors, the target point can only have an accurate position on one and only one occurrence of location of the horizon. In fact, each occurrence of location of the horizon gives rise to a new position of the target point and the union of a large number of these positions offers a distribution of the points of impact of the drilling well on the horizon.
In French patent application number 99 02 088 of Feb. 19, 1999, there is described a method for determining an optimal trajectory in order to reach a fuzzy target in a medium from a remote point. The method uses the geometrical uncertainties affecting a volume in order to determine an optimal trajectory for drilling. The volume is probabilized with the aid of the geometrical uncertainties and the optimal trajectory is that which best traverses the probabilized volume.
The subject of the present invention is a method making it possible to chart an impact point position probability map for a perfectly determined drilling well over a surface included within a three-dimensional medium with axes x, y, z and whose location is known with geometrical uncertainties. In particular, the said surface is a seismic horizon included within a seismic block defined in a three-dimensional reference frame with axes x, y, z and the location of the horizon is marred by uncertainties.
The method according to the invention relates to positioning risk. However, although its purpose is to chart risk maps, it does not do this by constructing successive migrations according to various velocities as in FR-A-2 763 702. More precisely, the method remedies the deficiencies of the method described in FR-A-2 763 702 by taking account of all the causes of uncertainties which affect the location of a horizon in the subsurface so as to chart an impact point position probability map.
Moreover, although the method according to the invention uses the geometrical uncertainties, its purpose is not to calculate an optimal trajectory as is described in French patent application number 99 02 088.
Referring to FIG. 7, the method, according to the invention, of producing impact point position probability for maps for a well over a surface S included within a three-dimensional medium and whose location is known with given geometrical uncertainties, is characterized in that it consists in:
defining a fixed, invariant target point chosen a priori on an initial location EO of the surface S (step 100),
discretizing the surface S with the aid of a grid composed of nodes and of grid cells (step 101),
assigning at least one elementary geometrical uncertainty vector to each node of the grid of the surface S (step 102),
determining a statistically significant number of occurrences of locations of the surface S as a function of the geometrical uncertainties affecting it (step 103),
projecting the target point onto each occurrence of location of the surface S so as to deduce a point of impact therefrom (step 104),
transferring the set of impact points onto the location EO of the surface S by allocating, to each of these points, surface co-ordinates identical to those which it had on the occurrence of location of the surface containing it (step 105),
defining over the surface S a statistical distribution for the set of impact points (step 106),
determining from the statistical distribution a probability density at all points of the surface S, the said density giving the probability that any point of the surface S is a point of impact (step 107), and
mapping over the location EO of the surface S the probability density as level curves (step 108).
According to a characteristic of the invention, the occurrences of locations of the surface are obtained by random drawings of the elementary uncertainty vectors.
According to another characteristic of the invention, several elementary geometrical uncertainty vectors are allocated at each node of the grid of the surface and a global uncertainty vector, which is the result of the said elementary geometrical uncertainty vectors, is calculated and is assigned to the relevant node.
According to another characteristic, the surface S being a seismic horizon defined by a pick and a depth migration, three elementary geometrical uncertainty vectors are allocated to each node of the grid of the seismic horizon, these being an uncertainty vector regarding the pick of the horizon, an uncertainty vector regarding the depth migration and an uncertainty vector regarding a seismic tie of the horizon to at least one drilling of the medium, so as to calculate the resultant vector of the said vectors.
According to another characteristic of the invention, at each node of the horizon, the elementary geometrical uncertainties vary in magnitude without varying in direction.
According to another characteristic of the invention, the variations in direction of the elementary geometrical uncertainties at each node of the horizon are predetermined.
According to another characteristic of the invention, at neighbouring nodes of the surface, the magnitudes of the resultant uncertainty vectors exhibit values such that the said nodes do not move independently of one another during the determination of each occurrence of location of the surface.
According to another characteristic of the invention, when the surface is traversed by at least one fracture, the points of contact of the surface with the fracture, as well as the part of the fracture plane which connects the pieces of the surface together are secured to the surface during the determination of each of its occurrences of location.
According to another characteristic of the invention, the determination of the probability density of the presence of any point of impact on the surface from the distribution of the impact points consists in:
inscribing the set of impact points within a quadrilateral standing on the surface S,
gridding the quadrilateral with the aid of grid cells all having the same area,
allocating each grid cell a probability value equal to the ratio of the number of impact points which traverse it to the total number of impact points inscribed within the quadrilateral.
In an advantageous implementation of the method according to the invention:
the target point C is defined as a point with co-ordinates Cx, Cy and Cz chosen a priori on an initial location of a surface in a three-dimensional medium,
the surface is fully gridded,
a set (n-tuple) of elementary geometrical uncertainty vectors is allocated to each node of the grid,
a random drawing of the magnitudes of the elementary uncertainty vectors is carried out at each node of the surface, then the resultant of the said vectors is formed and each node is moved according to the resultant uncertainty vector allocated to it, each set of new positions of the nodes defining a new occurrence of location of the surface,
the target point is projected vertically onto each occurrence of location of the surface so as to deduce therefrom a new point of impact,
each point of impact is transferred onto the initial location of the surface by allocating it, on this location, surface co-ordinates identical to those which it had on its own occurrence of location of surface,
a quadrilateral circumscribed about the set of impact points and standing on the surface is determined,
a secondary grid of the quadrilateral is constructed with the aid of secondary grid cells of mutually equal areas,
the number of impact points associated with each secondary grid cell is counted, which operation produces a distribution in the form of a histogram in the two dimensions defining the surface S,
a probability density is defined, on the basis of the histogram, in two dimensions by allocating each secondary grid cell a probability equal to the ratio of the number of impact points which reach it to the total number of impact points contained in the quadrilateral,
the probability of the presence of an impact point anywhere on the surface is calculated by using the two-dimensional probability density, and
impact point isoprobability curves are plotted on the surface